...the gain is relative to the square of the difference in focal length, not the linear difference:

If you double the linear dimension, you quadruple the area. That is simple arithmetic. But, as any mathematician or scientist knows: the resolution of a lens depends simply on the linear dimension - you get twice, not quadruple the resolution on doubling the focal length

Exactly. How is the spatial resolution of a dSLR lens generally specified? In line pairs per millimeter (LP/mm). That's a linear measurement, not the corresponding area measurement (mm2).

Jrista is correct in that the linear increase applies in both dimensions in terms of the area of the sensor covered with an increase in focal length, but AlanF is right on the physics and maths as pertains to resolution and S/N.

Of course you gain by going from 400 to 600mm - I have done it. Your arguments jrista are qualitative as to the amount of gain and are hand waving that it goes up by the focal length squared. But, the physics and maths quantitatively show it varies linearly with length. Let us just agree that it is a great improvement.

I went up to 600mm by using a 300mm f/2.8 II +2xTC III. It may not be quite as good as the native, but it is good enough and so light that I can hold it in my elderly hand for hours.

Here is a 100% crop of a heron 50-60 metres away I took on Sunday.

Perhaps an actual visual example from a world renown bird photographer can settle the argument. Art Morris, literally renown as the worlds best bird photographer, also agrees the gain is relative to the square of the difference in focal length, not the linear difference:

Support for an opinion may be comforting but doesn't prove or settle arguments. Art Morris often changes his mind. For many years, he much preferred the 400 f/5.6 (his "toy" lens) over the 100-400mm. Then he decided the zoom was better. For years, he argued against the 300mm f/2.8, then recently he changed his mind and decided it was great for bird photography.

I think all that is beside the point. Just look at the animated image...the bird clearly grows four times larger in area...not two times. It goes from covering about 20% of the frame to 80% of the frame. That is what I was trying to demonstrate. If you halve your angle of view, you halve it in both the horizontal and vertical...which means a 600mm lens covers 1/4 the scene area as a 300mm lens....if you shoot the same scene from the same physical spot with a 300mm lens and a 600mm lens, you could produce the same angular result with the 600 if you photographed the upper left, upper right, lower left, and lower right areas, and stitched them together.

That isn't a matter of opinion or personal preference...it's a matter of physics and math. Art Morris can't change his mind about that.

If you double the linear dimension, you quadruple the area. That is simple arithmetic. But, as any mathematician or scientist knows: the resolution of a lens depends simply on the linear dimension - you get twice, not quadruple the resolution on doubling the focal length; and the increase in precision or S/N depends on the square root of the area (number of pixels of the same sensor used) - you get twice the S/N, not quadruple. I can double the length and quadruple the area of a photo in Photoshop or with an enlarger and have the same size photo as produced by a lens of twice the focal length. By doing, so I get the same size image, so arguments based solely on image size are negated, but the S/N and resolution will both deteriorate by a factor of 2, not 4, compared with using the longer lens.

Those are the physics and maths. Now, show me the maths and physics that says otherwise, not hand waving. But, I will write no more about this subject.

I am not sure why you are talking at me about SNR. I said noise becomes a background factor, I did not say SNR changes by a factor of four anywhere in my posts so far. The SUBJECT grows in the frame by a factor of FOUR, fine detail grows by the same factor, which results in noise that was probably an issue when the subject only filled 20% of the frame no longer being an issue when the subject fills 80% of the frame. I never mentioned SNR specifically, because that really ultimately depends on how bright your subject is relative to the overall exposure (i.e. you may photograph a darker subject against midtoned or lighter surroundings...the SNR of the subject is still going to be relatively low), not it's size in the frame.

I would have like to see some photos in the review taken without flash, to get a true idea of how well the lens performs. Flash always make the images look artificially sharp, now matter how good (or bad) a lens is...

I would have like to see some photos in the review taken without flash, to get a true idea of how well the lens performs. Flash always make the images look artificially sharp, now matter how good (or bad) a lens is...

I am pretty sure that the sunset has been photographed without ... flash

I would have like to see some photos in the review taken without flash, to get a true idea of how well the lens performs. Flash always make the images look artificially sharp, now matter how good (or bad) a lens is...

The eagle shot is without flash. I was lucky to have a light cloud cover so the contrast was not too high and the whites and dark feathers were both exposed OK (1DX, ISO 1600, f4). Bald eagles in direct sunlight are tough. Processed only in DPP w some brightening to the shadows.